EXP 
EXP 
is the case with gunpowder ; but the expaiv 
sion of elastic fluids will burst solid sub- 
stances, 'and throw the fragments a great 
way off: for this two reasons have been 
assigned: 1. The immense velocity with 
which aerial fluids expand, when suddenly 
affected with high degrees of heat : and 2. 
The great celerity with which they acquire 
heat, and are affected by it. As an exam- 
ple, air when heated as much as iron, when 
brought to a white heat, is expanded to 
four times its bulk, but the metal itself will 
not be expanded the 500dth part of the 
space. In the case of gunpowder, which 
is well known as an explosive substance, 
the velocity with which the flame moves, 
is estimated at 7000 feet in a second. 
Hence the impulse of the fluid is inconceiv- 
ably great, and the obstacles on which it 
strikes are hurried off with vast velocity, 
viz. at the rate of 27 miles per minute. 
The velocity of the bullet is also promoted 
by the sudden propagation of the heat 
through the whole body of air, as soon as 
it is extricated from the materials of which 
the gunpowder is made, so that it strikes 
at once. Hence it has been inferred, that 
explosion depends first on the quantity of 
elastic fluid to be expanded : secondly, on 
the velocity it acquires by a certain degree 
of heat ; and thirdly, on the celerity with 
which the degree of heat affects the whole 
expansile fluid. 
EXPONENT, in algebra, is a number 
placed over any power or involved quan- 
tity, to shew to what height the root is 
raised : thus, 2 is the exponent of x 2 , and 4 
the exponent of x 4 , or xxxx. The rule for 
dividing powers of the same quantity, is 
to subtract the exponents, and make the 
difference the exponent of the quotient: 
if, therefore, a lesser power is divided by 
a greater, the exponent of the quotient 
must, by this rule, be negative : thus, 
a 4 • a 4 1 
— = a 4 - 6 = a . But — = — ; and 
a 6 a 6 a 2 
hence — is expressed by a 2 , with a negative 
a 
exponent. It is also obvious that — = 
a 
a} ~'=a°; but — = 1, and therefore a 0 =z 1. 
’ a 
, 1 a 0 
After the same manner, — — — = a 0 — 1 = 
a a 
a o _ 3 _ a _ s . s0 that the quantities, a, i ; 
1 1 1 1 „ 
— , —r.—r, — r, &c. may be expressed thus, 
a a 2 a 3 a' r 
a 1 , a 0 , a~', a -2 , a -3 , a -4 , &c. These 
are called the negative powers of a, which 
have negative exponents ; but they are at 
the same time positive powers of — or 
a 
a -1 . 
Exponent of a ratio, is the quotient aris- 
ing from the division of the antecedent by 
the consequent : thus, in the ratio of 5 to 
4, the exponent is li ; but the exponent of 
4 : 5, is £. If the consequent be unity, the 
antecedent itself is the exponent of the 
ratio : thus the exponent of the ratio 4 : 1 
is 4. Wherefore the exponent of a ratio is 
to unity as the antecedent is to the conse- 
quent. Although the quotient of the divi- 
sion of the antecedent by the consequent, 
is usually taken for the exponent of a ratio, 
yet in reality the exponent of a ratio ought 
to be a logarithm. And this seems to be 
more agreeable to Euclid's definition of 
duplicate and triplicate ratios, in his fifth 
book. For 1, 3, 9, are continual propor- 
tionals ; now if a be the exponent of the 
ratio of 1 to 3, and $ or r the exponent of 
the ratio of 3 to 9 ; and £ the exponent of 
the ratio of 1 to 9 j and since, according to 
Euclid, if three quantities be proportional, 
the ratio of the first to the third is said to 
be the duplicate of the ratio of the first to 
the second, and of the second to the third ; 
therefore according to tins, | must be the 
double of which is very false. But it is 
well known, the logarithm of the ratio of 1 
to 9, that is, the logarithm of 9, is the dou- 
ble of the ratio of 1 to 3, or 3 to 9, that is, 
the logarithm of 3. From whence it ap- 
pears that logarithms are more properly the 
exponents of ratios, than numerical quo- 
tients ; and Dr. Halley, Mr. Cotes, and 
others, are of the same opinion. 
Exponent, is also used in arithmetic, in 
the same sense as index or logarithm. 
EXPONENTIAL curve, is that whose 
nature is expressed by an exponential 
equation. The area of any exponential 
curve, whose nature is expressed by this 
exponential equation x* = y (making 1 -|-o 
= x) will be — - — n 2 -I — n 3 — 
‘ 0.1.2 ~ 0. 1.2.3 
1 _ . 1 1 
0.1. 2.3.4 V ‘ 0.1.2.3.4.5 ° ~ 0.1.2.3.4.5.6) 
V 6 , &c. 
Exponential equation, is that wherein 
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